Lecture 3 - Central limit Theorem

Central limit theorem

sum of a large number of independent random variables, each drawn from a probability distribution with finite mean and variance… will tend to be normally distributed, irrespective of the distribution function of the random variable

Central limit theorem images

Independence of Initial Distribution

regardless of original shape of the probability distribution (skewed, uniform, etc.) when you take the means of repeated samples, their distribution tends to resemble a normal distribution

fit between sample mean distribution and a normal distribution improves with…

larger sample sizes

Means of mean

sample mean is an unbiased estimator of the population mean

Reduction in spread

reduction by a factor of sqrtN… sample means tend to cluster more tightly around the true mean of the population

what can we assume from the CLT:

that the outcomes of measurements affected by multiple random factors follow a normal distribution, allowing us to apply various statistical tools that rely on Gaussian properties